Fractals and the second law of thermodynamics

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Online FractalAlex

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« on: June 04, 2020, 01:03:21 PM »
Let's say the Mandelbrot set is in a gigantic space of nothingness. It is an entire system of patterns and shapes. How much is the entropy in this fractal, and would the second law of thermodynamics apply there as well?

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Offline gerrit

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« Reply #1 on: June 04, 2020, 08:32:34 PM »
Let's say the Mandelbrot set is in a gigantic space of nothingness. It is an entire system of patterns and shapes. How much is the entropy in this fractal, and would the second law of thermodynamics apply there as well?
What does "entropy in a gigantic space of nothingness" mean to you? 2nd law will not apply as it refers to time and M-set is timeless.

Type "entropy" in the forum search box for some possible usages of this term in this context.


Online FractalAlex

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« Reply #2 on: June 04, 2020, 08:50:52 PM »
Well, entropy means that over time, a system gets more disordered, and the same thing goes for the universe. The Mandelbrot set has orbits that behave very chaotically, but is it a region of very high entropy? I don't know, if the term has a meaning or not. The second law of thermodynamics states that the total energy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. The same thing goes for the universe: entropy increases over time, and once it reaches thermodynamic equilibrium, heat death occurs, at a time frame of around \( 10^{1000} \) (if protons decay) to \( 10^{10^{120}} \) years (no proton decay). This fractal has some order of disorder/chaos, which means if it were in a 3D or 4D space in our plane of existence, the structures would more or less have some level of entropy, but I'm not sure if this is how it would work...

Offline gerrit

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« Reply #3 on: June 05, 2020, 09:19:16 AM »
Entropy of a thing is the log of the number of microstates that can realize it.
If the thing is the Mandelbrot set what would the microstates be?

Maybe the information theoretical entropy (Shannon) has some application to what you want?

Offline gerrit

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« Reply #4 on: June 05, 2020, 10:27:14 AM »
https://en.wikipedia.org/wiki/Approximate_entropy looks perhaps interesting.
It gives a recipe to compute "r-entropy" which can be applied to an orbit which perhaps could be visualized and may look good.
I don't look forward to coding that in Ultra Fractal, maybe I'll try in Matlab or hopefully someone will beat me to it.

Offline gerrit

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« Reply #5 on: June 06, 2020, 10:57:11 PM »
Tried it but it doesn't look particularly good and is super slow. Main problem is discontinuities making for ugly pictures. A little better for non-escaping orbits as we compare obits of same length. Unless there's a way to speed it up and smooth it not useful I think.
Code: [Select]
function apen = ApEn( dim, r, data, tau )
%ApEn
%   dim : embedded dimension
%   r : tolerance (typically 0.2 * std)
%   data : time-series data
%   tau : delay time for downsampling
%   Changes in version 1
%       Ver 0 had a minor error in the final step of calculating ApEn
%       because it took logarithm after summation of phi's.
%       In Ver 1, I restored the definition according to original paper's
%       definition, to be consistent with most of the work in the
%       literature. Note that this definition won't work for Sample
%       Entropy which doesn't count self-matching case, because the count
%       can be zero and logarithm can fail.
%
%       A new parameter tau is added in the input argument list, so the users
%       can apply ApEn on downsampled data by skipping by tau.
%---------------------------------------------------------------------
% coded by Kijoon Lee,  kjlee@ntu.edu.sg
% Ver 0 : Aug 4th, 2011
% Ver 1 : Mar 21st, 2012
%---------------------------------------------------------------------
if nargin < 4, tau = 1; end
if tau > 1, data = downsample(data, tau); end
   
N = length(data);
result = zeros(1,2);
for j = 1:2
    m = dim+j-1;
    phi = zeros(1,N-m+1);
    dataMat = zeros(m,N-m+1);
   
    % setting up data matrix
    for i = 1:m
        dataMat(i,:) = data(i:N-m+i);
    end
   
    % counting similar patterns using distance calculation
    for i = 1:N-m+1
        tempMat = abs(dataMat - repmat(dataMat(:,i),1,N-m+1));
        boolMat = any( (tempMat > r),1);
        phi(i) = sum(~boolMat)/(N-m+1);
    end
   
    % summing over the counts
    result(j) = sum(log(phi))/(N-m+1);
end
apen = result(1)-result(2);
end


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